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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Dynamics of macro-observables and space-time inhomogeneous gibbs ensembles

L. Lanz, G. Lupieri

Published Oct 1, 1978• 3 citations
Publisher
DOI
10.1007/bf02894589

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arXiv:2412.01896•Physical Review A•May 12, 2025•4 citations
We study the large time dynamics of a macroscopically large quantum system under a sudden quench. We show, first of all, that for a generic system in the thermodynamic limit the Gibbs distribution correctly captures the large time dynamics of its global observables. In contrast, for an integrable system, the generalized Gibbs ensemble captures its global large time dynamics only if the system can be thought of as a number of noninteracting uncorrelated fermionic degrees of freedom. The conditions for the generalized Gibbs ensemble to capture the large time dynamics of local quantities are likely to be far less restrictive, but this question is not systematically addressed here.
In recent years, statistical physics tools have been used to study the dynamics of macroscopic quantum and classical integrable models. These studies have shown that, in the thermodynamic limit, long term time-averages of local observables equal statistical averages calculated with a Generalized Gibbs Ensemble (GGE). In the latter, the weight in phase space involves all constants of motion multiplied by Lagrange multipliers which are fixed by their initial values. We argue that the GGE can be interpreted as a proper Boltzmann measure of a different (non-standard) mechanical model. This measure can therefore be sampled with stochastic methods implemented, for example, as a Langevin process. We tailor the deterministic force in the Langevin equations so that at long times the stochastic trajectories sample the desired measure. We use the integrable Neumann Model -- the model of a classical particle constrained to move on a sphere under anisotropic harmonic potentials -- as a benchmark. We test the method in a setup where the Lagrange multipliers are fixed by the GGE governing the asymptotic dynamics of the isolated Neumann model following a quench. This problem has different phases depending on the initial conditions and quench protocol and, in all cases, with the stochastic evolution of an adequately tuned Langevin process we reproduce the asymptotic values of relevant observables.
Unitary integrable models typically relax to a stationary Generalized Gibbs Ensemble (GGE), but in experimental realizations dissipation often breaks integrability. In this work, we use the recently introduced time-dependent GGE (t-GGE) approach to describe the open dynamics of a gas of bosons subject to atom losses and gains. We employ tensor network methods to provide numerical evidence of the exactness of the t-GGE in the limit of adiabatic dissipation, and of its accuracy in the regime of weak but finite dissipation. That accuracy is tested for two-point functions via the rapidity distribution, and for more complicated correlations through a non-Gaussianity measure. We combine this description with Generalized Hydrodynamics and we show that it correctly captures transport at the Euler scale. Our results demonstrate that the t-GGE approach is robust in both homogeneous and inhomogeneous settings.